Lemma 76.42.8 (08BB)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.42: Grothendieck's existence theorem
Lemma 76.42.8 (cite)

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Lemma 76.42.8. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$ -module, $(\mathcal{F}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$ , and $\alpha : (\mathcal{F}_ n) \to \mathcal{G}^\wedge$ a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$ . Then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where

$\mathcal{F}$ is a coherent $\mathcal{O}_ X$ -module,
$a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$ -module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$ ,
$\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge$ is an isomorphism, and
$\alpha = a^\wedge \circ \beta$ .

Proof. The uniqueness and étale descent for objects of $\textit{Coh}(X, \mathcal{I})$ and $\textit{Coh}(\mathcal{O}_ X)$ implies it suffices to construct $(\mathcal{F}, a, \beta )$ étale locally on $X$ . Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.6. $\square$

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